Formula Sheet - Differential Equations Formula Sheet µ (...

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Unformatted text preview: Differential Equations Formula Sheet µ ( x) = e ∫ M y −Nx N dx µ ( y) = e ∫ N x −M y M dT = k ( T − Tm ) dt dy P (t ) = P0 K P0 + ( K − P0 ) e − r t d 2x + ω 2 x = 0 , ω 2 = k / m , x(t ) = c1 cos ω t + c 2 sin ω t , T = 2π / ω , f = 1 / T = ω / 2π dt 2 x(t ) = c1 cos ω t + c 2 sin ω t = A sin ( ω t + φ ) , A = c1 2 + c 2 2 , sin φ = c1 c , cos φ = 2 A A β k d 2x dx 2 + 2λ + ω 2 x = 0 , 2λ = , ω = 2 m m dt dt − λt λ −ω t + c2 e − λ2 − ω 2 > 0 , x(t ) = e c1e − λt 2 2 λ − ω = 0 , x(t ) = e ( c1 + c 2 t ) 2 2 λ 2 −ω 2 t ( λ2 − ω 2 < 0 , x(t ) = e − λt c1 cos ω 2 − λ2 t + c 2 sin ω 2 − λ2 t Ends of the Beam Embedded Free Simply Supported 1, x, x 2 , , x n −1 ⇒ ) Boundary Conditions y = 0, y ′ = 0 y ′′ = 0, y ′′′ = 0 y = 0, y ′′ = 0 EI d4y = w( x) dx 4 Dn eαx , xeαx , x 2 eαx , , x n −1eαx ⇒ ( D − α ) eαx cos βx, xeαx cos βx, x 2 eαx cos βx, , x n−1eαx cos βx 2 2 2 ⇒ D − 2αD + α + β αx αx 2 αx n −1 αx e sin βx, xe sin βx, x e sin βx, , x e sin βx n [ y = c1 x m1 + c 2 x m2 y = c1 x m1 + c 2 x m1 ln x + c3 x m1 (ln x) 2 + + c k x m1 (ln x) k −1 y = x α [ c1 cos( β ln x ) + c 2 sin ( β ln x ) ] ( )] n ...
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This note was uploaded on 01/13/2012 for the course MATH 2914 taught by Professor Pamelasatterfield during the Fall '10 term at NorthWest Arkansas Community College.

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